3.The position of nine trees which are to be planted a long the sides of a road, five on the north side and four on the south side.
If there are 3 figs, 4 prunes and 2 magnolias, find the number of different ways in which these could be planted assuming that the trees of the same species are identical.

Jan 24th 2010, 06:11 AM

HallsofIvy

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Originally Posted by liptonpc

Hi all,
i have find this exercise in internet and try solve it but cant, pls help me.
Thanks

1.Given A = 3.9 after rounding. Find the range of real value for A before rounding if A has been round to
(i) 1 decimal places.
(ii) 2 significant figures

Here, "1 decimal place" and "2 significant figures" are the same thing! Any number greater than or equal 3.85 and less than 3.95 will be rounded to 3.9. Here, I have used the convention that "if the first digit to be dropped is 5, round up" (since any digits after that will make it close to the larger number) but some people use the convention "if the first digit to be dropped is 5, round to the nearest even digit" (that way you will be rounding up half the time and down half the time so the errors will tend to cancel). Using that convention, since "9" is odd, any number greater than 3.85 and less than 3.95 will be rounded to 3.9.

The largest these number can be, and still be rounded to 2.0 and 3.0, are 2.5 and 3.5. 2x+ y, in that case, would be 5.0+ 3.5= 8.5. The smallest they can be are 1.5 and 2.5. In that case, 2x+ y would be 5.5. 2(2.0)+ 3.0= 7. Since 8.5-7= 1.5 and 7- 5.5= 1.5, the "absolute error is 1.5. The "relative error" is 1.5/7= 3/14 or about .214.

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(ii) $\displaystyle \frac{x+z}{y}$

Now, try this one yourself.

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3.The position of nine trees which are to be planted a long the sides of a road, five on the north side and four on the south side.
If there are 3 figs, 4 prunes and 2 magnolias, find the number of different ways in which these could be planted assuming that the trees of the same species are identical.

Since there are no conditions on which can be planted on the north side and which on south, we can really ignore that distinction. This is really about planting 9 trees in 9 places. If they were all different, there would be, of course, 9! diferent ways of doing that. But that includes cases in which two figs, say were switched while all other trees remainded in the same place and we don't want to count that as two different ways. Since there are 3 figs, there are 3! ways to interchange them while leaving the other trees the same- for every way of planting the trees, we have counted 3! that are really the same since they only involve swapping fig trees. That means that the number of ways of planting the trees if we don't count those as different is 9!/3!. Of course, we can treat the prunes and magnolias the same way- the number of distinct ways of planting these trees is 9!/(3!4!2!)